Problem: Solve for $x$ and $y$ by deriving an expression for $x$ from the second equation, and substituting it back into the first equation. $\begin{align*}-2x-6y &= 9 \\ -x+2y &= 1\end{align*}$
Begin by moving the $y$ -term in the second equation to the right side of the equation. $-x = -2y+1$ Divide both sides by $-1$ to isolate $x$ $x = {2y - 1}$ Substitute this expression for $x$ in the first equation. $-2({2y - 1}) - 6y = 9$ $-4y + 2 - 6y = 9$ Simplify by combining terms, then solve for $y$ $-10y + 2 = 9$ $-10y = 7$ $y = -\dfrac{7}{10}$ Substitute $-\dfrac{7}{10}$ for $y$ in the top equation. $-2x-6( -\dfrac{7}{10}) = 9$ $-2x+\dfrac{21}{5} = 9$ $-2x = \dfrac{24}{5}$ $x = -\dfrac{12}{5}$ The solution is $\enspace x = -\dfrac{12}{5}, \enspace y = -\dfrac{7}{10}$.